I was told that, by my assumption that F was to be a homomorphism and how I then built the mapping from there, I was only sufficiently showing that the mapping was a homomorphism for those specific cases. Not every pairing of elements from the domain have been proven to be homomorphic.

The answer is not wrong, really, it's more of an argument of weather this method can be proven. The only other way I was told this could be done was through a guess and check process. I believe this algorithm works.....